How to Write a Proof — Challenge
A proof is not a calculation. It is a chain of statements where every link carries a reason.
What these problems cover
These questions use 17 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Perimeter — Distance all the way around the outside.
- Base — The number being multiplied. In 53 the base is 5.
- Scale — What one grid step is worth — and where the axis starts.
- Converse — If a2 + b2 = c2, then the triangle must be right-angled.
- Congruent — Same size and same shape. Written △ ABC ≅ △ DEF.
- Transversal — A line that crosses two or more other lines.
- Corresponding angles — Same position at each crossing. Equal when the lines are parallel.
- Alternate interior angles — Between the parallels, on opposite sides of the transversal. Equal.
- Midpoint — The point halfway between two others: average the x values and average the y values.
- Parallel — Same slope, different intercept.
- Segment — Two endpoints and everything between them. Written AB.
- Prove — The statement you must arrive at. Always the last line.
- Two-column proof — Statements on the left, the reason for each on the right.
- Counterexample — A single case that makes a claim false. One is enough.
- CPCTC — Corresponding Parts of Congruent Triangles are Congruent — used after proving congruence.
- Reflexive property — Anything is congruent to itself. A shared side is congruent to itself.
- Rhombus — A parallelogram with four congruent sides.
Problem 1
What is a statement accepted without proof called?
- An axiom
- A theorem
- A corollary
- A conjecture
- A proof
Worked answer
An axiom
Every system of reasoning has to start somewhere, with statements taken as given.
Those starting statements are axioms, or postulates.
A theorem, by contrast, is something established by argument from the axioms.
Problem 2
What is a statement proved from axioms called?
- An axiom
- A theorem
- A definition
- A guess
- An example
Worked answer
A theorem
A theorem is a statement shown to be true by a chain of reasoning from the axioms.
A definition names an idea rather than asserting anything that needs proving.
A guess with no proof yet is a conjecture.
Problem 3
Two lines cross. What is true of the vertical angles formed?
- They are supplementary
- They are equal
- They are complementary
- They are unequal
- They are right angles
Worked answer
They are equal
Each of the two vertical angles is on a straight line with the same neighbouring angle.
So each equals 180 degrees minus that neighbour, which makes them equal to each other.
Adjacent angles at the crossing are the ones that are supplementary.
Problem 4
A transversal crosses two parallel lines. What is true of alternate interior angles?
- They are supplementary
- They are complementary
- They are equal
- They are unequal
- They are reflex
Worked answer
They are equal
Corresponding angles at parallel lines are equal, and vertical angles are equal.
Combining those two facts shows that alternate interior angles are equal as well.
Co-interior angles, on the same side of the transversal, are the supplementary pair.
Problem 5
What does a single counterexample do to a general claim?
- Proves it
- Disproves it
- Strengthens it
- Has no effect on it
- Proves its converse
Worked answer
Disproves it
A general claim asserts something about every case.
One case where it fails is enough to show the claim is false as stated.
The asymmetry is worth noting: no number of confirming examples can prove a general claim, but one counterexample can refute it.
Problem 6
What is the converse of the statement "if P then Q"?
- If Q then P
- If not P then not Q
- If not Q then not P
- P and Q
- Not P
Worked answer
If Q then P
The converse swaps the two parts, giving "if Q then P".
A statement and its converse are different claims: either may be true without the other.
"If it is a square then it is a rectangle" is true, but its converse is not.
Problem 7
What is the contrapositive of the statement "if P then Q"?
- If Q then P
- If not P then not Q
- If not Q then not P
- P and Q
- Not P
Worked answer
If not Q then not P
The contrapositive negates both parts and swaps them, giving "if not Q then not P".
Unlike the converse, the contrapositive is always true exactly when the original is, so proving one proves the other.
That equivalence is what makes proof by contrapositive a valid method.
Problem 8
In a proof by contradiction, what is assumed at the start?
- The conclusion
- The opposite of the conclusion
- Nothing at all
- That the axioms are false
- The converse
Worked answer
The opposite of the conclusion
The method begins by supposing the statement to be proved is false.
Reasoning from that supposition until something impossible appears shows the supposition cannot hold.
The statement must therefore be true. Assuming the conclusion itself would be circular reasoning.